2018/07/31 by Tamara Kohler, Toby Cubitt · 8 citations
Computer Science · Mathematics · Physics and Astronomy · #Hamiltonian (control theory) #Invariant (physics) #Lattice (music) #Markov Chains and Monte Carlo Methods #Quantum Computing Algorithms and Architecture #Quantum many-body systems #Simple (philosophy) #Spin model #cond-mat.stat-mech #quant-ph
paper · pdf · doi:10.1007/s10955-019-02295-3
published in Journal of Statistical Physics 176(1), 228-261 (Springer Science+Business Media) · 38 pages, 3 figures. v2: Extended discussion, simplified overview
openalex created_date 2018/07/10 · openalex publication_date 2019/04/30 · arxiv created 2019/05/02 · arxiv updated 2019/05/22 · openalex updated_date 2026/08/05
Spin models are widely studied in the natural sciences, from investigating magnetic materials in condensed matter physics to studying neural networks. Previous work has demonstrated that there exist simple classical spin models that are universal: they can replicate—in a precise and rigorous sense—the complete physics of any other classical spin model, to any desired accuracy. However, all previously known universal models break translational invariance. In this paper we show that there exist translationally invariant universal models. Our main result is an explicit construction of a translationally invariant, 2D, nearest-neighbour, universal classical Hamiltonian with a single free parameter. The proof draws on techniques from theoretical computer science, in particular recent complexity theoretic results on tiling problems. Our results imply that there exists a single Hamiltonian which encompasses all classical spin physics, just by tuning a single parameter and varying the size of the lattice. We also prove that our construction is optimal in terms of the number of parameters in the Hamiltonian; there cannot exist a translationally invariant universal Hamiltonian with only the lattice size as a parameter.